**1. Introduction**

The most common problem in boundary layer flow that had been treated so far is much focused on those for steady flows. Even though it is the steady flows that seem to have the utmost significance in real-world demands, some cases of time-varying in the boundary layer which is unsteady indicate an important role in several engineering problems. Some of the examples are start-up processes where the motions in rest are transits from a steady flow to another, and periodic motions of the working fluid [1]. The behavior of unsteady boundary layer flow describes an unusual pattern compared to the steady flow owing to the additional time-dependent terms in the governing equations, which exaggerated the separation of boundary layer and the fluid motion arrangemen<sup>t</sup> [2,3]. The thermal and mechanical properties of such an unsteady mechanism in the boundary layer approximation have been studied both analytically and numerically. Elbashbeshy and Bazid [4] presented the numerical investigation towards the unsteady stretching surface with heat transfer analysis. At the same time, Bhattacharya [5]

managed to prove the existence of dual solutions in unsteady stagnation point flow towards a shrinking sheet by employing the shooting method approach coupled with a Runge–Kutta integration scheme. Bachok et al. [6] concluded that inclusion of the unsteadiness parameter o ffers a significant impact towards the boundary layer flow in nanofluid and Fan et al. [7] presented analytical solutions using the homotopy analysis method (HAM) and managed to advertise a highly precise analytical estimation which is in excellent agreemen<sup>t</sup> with the numerical results o ffered by the Keller box scheme. It is worth mentioning that a considerable amount of reviews on the unsteady stagnation point flow due to a stretching/shrinking surface have been accomplished by numerous researchers, including [8–11].

The stagnation point flow is one of the important topics in mechanics of fluid, in the way that stagnation point generally occurs in both engineering and science flow fields. The stagnation point flow could be identified in the extrusion process, polymer industry, and plane counter jet [12,13]. The ground-breaking research in this topic was first initiated by Hiemenz [14] who exposed an analytical explanation of two-dimensional stagnation point flow, and soon after, Homann [15] conducted a classical study of stagnation point in three-dimensional flow with regard to an axisymmetric case; whereas Howarth [16] tackled the problem of non-axisymmetric flow close to the stagnation area in three-dimensional analysis. Recently, Khashi'ie et al. [17,18], Fang and Wang [19], Waini et al. [20], and Zainal et al. [21] have scrutinized the stagnation point flow problems in diverse aspects with no-slip boundary conditions. Nevertheless, in numerous engineering occasions, the slip e ffect should be comprised, such as flow over lubricated or coated surfaces, rough or striated surfaces [22] and internal rare field gas flow [23]. Examples of industrial applications involving the slip boundary conditions are fluid flow on multiple interfaces, rare field fluid problems, and also the reacting flow in reactors [24,25]. Navier [26] and Maxwell [27] were the primary researchers who pioneered the study of linear slip boundary conditions, while Wang [28] has well reflected a comprehensive theoretical analysis considering the no-slip boundaries concentrating on the stagnation point flow. Rao and Rajagopal [29] have conducted an extensive evaluation and argumen<sup>t</sup> between the slip and no-slip condition, and Jusoh et al. [30] deliberated a modified nanofluid model towards a stretching/shrinking surface by considering a velocity slip parameter in three-dimensional flow. The study revealed that an increase in the velocity slip magnitude contributed to the intensification of skin friction coe fficients.

In certain cases, the velocity slip or the non-adherence of the fluid to a solid boundary phenomenon was witnessed, for instance, in the micro-scale devices [31]. Fluids promoting slip are critical in technical applications such as polishing artificial heart valves and internal cavities [32]. The flow behavior and the shear stress in the fluid are rather distinctive with a slip at the wall boundary compared to those with no-slip condition. Besides that, the velocity slip e ffect does influence the heat transfer rate and was confirmed by Mukhopadhyay [33], who had investigated the slip impact of the unsteady mixed convective flow towards a porous stretching surface with heat transfer. The analysis found that the heat transfer rate declines with the velocity slip parameter, while it upsurges with the unsteadiness parameter. Mahapatra and Nandy [34] conducted a numerical study of the unsteady stagnation point flow past a shrinking sheet and heat transfer with the presence of slip e ffects in a viscous fluid. The results conveyed that with the increase of the velocity slip and unsteady parameter, the heat transfer rate is reported to escalate. Meanwhile, in nanofluid flow, Majumder et al. [35] specified that exertion of the partial velocity slip against the sheet surface is common. By relying on the finding in [35], Noghrehabadi et al. [36] then examined the impact of partial velocity slip on the nanofluid boundary layer flow and heat transfer past a stretching sheet. The work in [36] reported that an increment in the velocity slip e ffect decreases the momentum boundary layer thickness. Van Gorder et al. [37], who examined the nanofluid boundary layer flow over a stretching surface, also conveyed a similar result as [36] and further explained that no-slip condition is not applicable for fluid flows at nanoscales. Besides that, Dinarvand and Rostami [38] studied the rotating nanofluid flow and heat transfer with the presence of internal heating, velocity slip, and di fferent shapes of nanoparticles. They showed that an increment in the velocity slip e ffect reduces the skin friction coe fficient significantly. Researchers also tend to analyze the e ffect of velocity slip in the unsteady nanofluid flow as unsteady flow problems

are more relatable to real-world applications. For instance, Seth et al. [39] studied the unsteady hydromagnetic nanofluid flow and heat transfer past a non-linearly stretching surface with Navier's velocity slip and presented the analysis of entropy generation. Other valuable references regarding unsteady nanofluid flow with the velocity slip effect can be found in [40–42].

Ever since the outstanding inventions achieved by Choi and Eastman [43], who originated the brilliant idea of demonstrating the nanoparticle suspension in a base fluid and came out with the nanofluid term, a better type of working fluid is still being pursued. Acknowledging the sufficient improvement in the thermal conductivity of the conventional fluid is crucial, an advanced nanofluid form known as hybrid nanofluid is introduced, which intends to have highly developed heat conductivity. This modern type of fluid agen<sup>t</sup> has fascinated numerous researchers owing to its reputation in the emergence and improvement of thermal characteristics in realistic applications, including micro-channel, heat pipes, heat exchangers, air conditioning systems, and mini-channel heat sink [44,45]. Gupta et al. [46] and Xian et al. [47] have reviewed the preparation method of hybrid nanoparticles along with the stabilization and its significance in industrial sectors. One of the critical elements in establishing a sustainable hybrid nanofluid suspension is selecting an appropriate combination of nanoparticles. The most widely used nanoparticles for the formation of hybrid nanofluid suspension are carbon materials (graphite, MWCNTs, CNTs), metals (Cu, Ag), metal oxides (Al2O3, CuO, Fe2O3), metal carbide, and a metal nitride. Madhesh and Kalaiselvam [48] conducted an experimental study to examine the features of hybrid nanofluid as a coolant agent, Tahat and Benim [49] examined the efficiency of hybrid nanofluid on flat plate solar collector, and they verified that the viscosity, thermal conductivity, and density of the working fluid had increased together with the concentration of Al2O3/CuO concentration, thus enhancing the solar collector proficiency. Some early research on hybrid nanofluid that employed the numerical method was done by Labib et al. [50], who investigated the impact of base fluids and hybrid nanofluid using a two-phase mixture model in forced convective heat transfer. Moghadassi et al. [51] revealed that the heat transfer performance is enhanced by adding the nanoparticles of Al2O3-Cu hybrid nanofluid while creating a small pressure drop in the system regime, Devi and Devi [52] focused on the mathematical inspection towards a stretching sheet. In contrast, the evaluation of heat transfers in the natural convection of Al2O3/water nanofluid and Al2O3-Cu/water hybrid nanofluid with a discrete heat source was explored by Takabi and Salehi [53]. Additional details on this topic are well described in the literatures [54–56].

To the best of the authors' knowledge, the existing literature does not consider the unsteady stagnation point flow of hybrid nanofluids with the presence of velocity slip parameter and stability analysis in their models. Thus, the addressed issues above have inspired the authors to perform a numerical study in unsteady stagnation point flow towards a convectively heated stretching/shrinking sheet in alumina–copper/water (Al2O3-Cu/H2O) with the impact of velocity slip on heat transfer. The hybrid nanofluid is recognized by dispersing Al2O3 nanoparticles into H2O, followed by Cu with different volume fractions and the thermophysical properties of the hybrid nanofluid are adopted from Ghalambaz et al. [57] and Takabi and Salehi [53], which were based on the feasible physical assumptions and are in agreemen<sup>t</sup> with the conservation of mass and energy. The present work also utilized the bvp4c approach, which can be accessed in the MATLAB programming system towards solving the formulated problem. The existence of more than one solution is predictable; thus, an analysis of solution stability is completed to confirm the steadiness of the solutions which has an actual physical interpretation. The explanation of the results and the convergence of the obtained solutions are deliberated on in detail. Particular cases of current findings are evaluated in accordance with those of Mahapatra and Nandy [34] and Wang [58]. Furthermore, the consensus between previous and current findings is outstanding, and the agreemen<sup>t</sup> is excellent.
